This is true in all dimensions, and can be proved by induction (on $d$) applied to the following (slightly stronger) hypothesis:

**Theorem:** If $P$ is a convex $d$-polytope with $k$-in-spheres for all $k \in [0, d-1]$, then:

- $P$ is regular.
- $P$ is determined (up to an element of the orthogonal group $O(d)$) by that $d$-tuple $(r_0, r_1, \dots, r_{d-1})$ of $k$-in-radii.
- $P$ is determined completely if additionally a facet (codimension-1 face) $Q$ of $P$ is specified.

**Proof:** If the polytope $P$ has squared $k$-in-radii $(r_0^2, r_1^2, \dots, r_{d-1}^2)$, then every facet of $P$ has squared $k$-in-radii $(r_0^2 - r_{d-1}^2, r_1^2 - r_{d-1}^2, \dots, r_{d-2}^2 - r_{d-1}^2)$. By the first two parts of the inductive hypothesis, all facets of $P$ are therefore regular and congruent to each other (being determined by these $k$-in-radii).

Now, given a facet $Q$ of $P$ and a facet $R$ of $Q$, let $\Pi$ be the hyperplane through the origin which contains $R$. Let $Q'$ be the other facet of $P$ which contains $R$. Because the $k$-in-spheres of $Q'$ are the reflections (in $\Pi$) of the $k$-in-spheres of $Q$, and they share a common facet $R$, it follows (from the third part of the inductive hypothesis) that $Q'$ is the reflection of $Q$ through the hyperplane $\Pi$.

As the boundary $\partial P$ (i.e. the union of all facets) is homeomorphic to $S^{d-1}$, we can reach any facet $Q_1$ from any facet $Q_0$ by a 'path' of 'adjacent' (i.e. sharing a common subfacet) facets. Consequently, we can transform any facet into any other facet by a sequence of reflections in hyperplanes through the origin. As each facet is flag-transitive, it therefore follows that $P$ is flag-transitive (i.e. regular) as desired.

Moreover, this reflection procedure of building $P$ from a single facet $Q$ establishes the third part of the theorem.

This leaves the second part of the theorem. Suppose $P$ and $P'$ are two polytopes sharing the same set of $k$-in-spheres. Let $Q$ be an arbitrary facet of $P$, and $Q'$ be an arbitrary facet of $P'$. By the inductive hypothesis, $Q$ and $Q'$ are congruent; let $f$ be an isometry of the ambient space which maps $Q$ to $Q'$. The origin is either mapped to itself or (if we chose the 'wrong' isometry) to $2v$, where $v$ is the centroid of $Q$; we can if necessary reflect again in the hyperplane containing $Q$ to ensure the origin is preserved by $f$. Consequently, $f$ is an element of the orthogonal group $O(d)$ which maps $Q$ to $Q'$. By the third part of the theorem (which we've already proved), $f$ must map $P$ to $P'$, establishing the second part of the theorem.

assumethat the spheres are concentric, or do you say that theymust beconcentric? $\endgroup$ – Wlodek Kuperberg Jun 1 '20 at 19:46